Maxwell's equations constitute first-order coupled partial differential equations. They can be uncoupled at the expense of raising the order of the differentiation.  This solution is know as the wave equation.  In order to demonstrate the procedure of deriving the wave equation, consider Maxwell's equations with losses and with constant constitutive parameters. 
\begin{align}
    \nabla\cdot\vec{\mathcal{E}}&=\frac{\rho_{ev}}{\varepsilon}&&\text{Gauss's Electric Field Law}\label{eqn:geflsf}\\
    \nabla\cdot\vec{\mathcal{H}}&=\frac{\rho_{mv}}{\mu}&&\text{Gauss's Magnetic Field Law}\label{eqn:gmflsf}\\
    \nabla\times\;\vec{\mathcal{E}}&=-\mu\frac{\partial\vec{\mathcal{H}}}{\partial{t}}-\sigma_m\vec{\mathcal{H}}-\mathcal{\vec{M}}_i&&\text{Faraday's Law}&&\label{eqn:flsf}\\
    \nabla\times\,\vec{\mathcal{H}}&=\varepsilon\frac{\partial\vec{\mathcal{E}}}{\partial{t}}+\sigma_e\vec{\mathcal{E}}+\mathcal{\vec{J}}_i&&\text{Ampere's Law}\label{eqn:alsf}
\end{align}
Taking the curl of both sides of (\ref{eqn:flsf}) yields,
\begin{align}
\nabla\times\nabla\times\vec{\mathcal{E}}&=-\left(\mu\frac{\partial}{\partial{t}}+\sigma_m\right)\nabla\times\vec{\mathcal{H}}-\nabla\times\vec{\mathcal{M}_i}\label{eqn:curlflsf}
\end{align}
Using the vector identity $\nabla\times\nabla\times\vec{A} = -\nabla^2\vec{A}+\nabla(\nabla\cdot\vec{A})$ and substituting (\ref{eqn:alsf}) into (\ref{eqn:curlflsf}) gives,
\begin{align}
\nabla^2\vec{\mathcal{E}}-\nabla(\nabla\cdot\vec{\mathcal{E}})&=\left(\mu\frac{\partial}{\partial{t}}+\sigma_m\right)\left(\varepsilon\frac{\partial\vec{\mathcal{E}}}{\partial{t}}+\sigma_e\vec{\mathcal{E}}+\vec{\mathcal{J}}_i\right)+\nabla\times\vec{\mathcal{M}}_i\label{eqn:waveeqn1}
\end{align}
Rearranging and substituting (\ref{eqn:geflsf}) into (\ref{eqn:waveeqn1}) gives the following wave equation,
\begin{align}
\nabla^2\vec{\mathcal{E}}-\mu\varepsilon\frac{\partial^2\vec{\mathcal{E}}}{\partial{t}^2}-\left(\mu\sigma_e+\varepsilon\sigma_m\right)\frac{\partial\vec{\mathcal{E}}}{\partial{t}}-\sigma_m\sigma_e\vec{\mathcal{E}}=\mu\frac{\partial\vec{\mathcal{J}}_i}{\partial{t}}+\sigma_m\vec{\mathcal{J}}_i+\nabla\times\vec{\mathcal{M}_i}+\frac{\nabla\rho_{ev}}{\varepsilon}\label{eqn:waveeqn2}
\end{align}
By following a similar procedure a wave equation for the magnetic field intensity can also be derived.  
\begin{align}
\begin{split}
\nabla^2\vec{\mathcal{H}}-\mu\varepsilon\frac{\partial^2\vec{\mathcal{H}}}{\partial{t}^2}-\left(\mu\sigma_e+\varepsilon\sigma_m\right)\frac{\partial\vec{\mathcal{H}}}{\partial{t}}-\sigma_m\sigma_e\vec{\mathcal{H}}\\=\varepsilon\frac{\partial\vec{\mathcal{M}}_i}{\partial{t}}+\sigma_e\vec{\mathcal{M}}_i-\nabla\times\vec{\mathcal{J}_i}+\frac{\nabla\rho_{mv}}{\mu}
\end{split}\label{eqn:waveeqn3}
\end{align}
When there are no sources $\vec{\mathcal{M}_i}=\vec{\mathcal{J}_i}=\rho_{mv}=\rho_{ev}=0$, then (\ref{eqn:waveeqn2}) and (\ref{eqn:waveeqn3}) reduce too,
\begin{align}
\nabla^2\vec{\mathcal{E}}&=\mu\varepsilon\frac{\partial^2\vec{\mathcal{E}}}{\partial{t}^2}+\left(\mu\sigma_e+\varepsilon\sigma_m\right)\frac{\partial\vec{\mathcal{E}}}{\partial{t}}+\sigma_m\sigma_e\vec{\mathcal{E}}\label{eqn:waveeqn4}\\
\nabla^2\vec{\mathcal{H}}&=\mu\varepsilon\frac{\partial^2\vec{\mathcal{H}}}{\partial{t}^2}+\left(\mu\sigma_e+\varepsilon\sigma_m\right)\frac{\partial\vec{\mathcal{H}}}{\partial{t}}+\sigma_m\sigma_e\vec{\mathcal{H}}\label{eqn:waveeqn5}
\end{align}
If there are no losses, then (\ref{eqn:waveeqn4}) and (\ref{eqn:waveeqn5}) reduce too,
\begin{align}
\nabla^2\vec{\mathcal{E}}&=\mu\varepsilon\frac{\partial^2\vec{\mathcal{E}}}{\partial{t}^2}\label{eqn:waveeqn6}\\
\nabla^2\vec{\mathcal{H}}&=\mu\varepsilon\frac{\partial^2\vec{\mathcal{H}}}{\partial{t}^2}\label{eqn:waveeqn7}
\end{align}
In order to keep the discussion general, a generic field quantity of $\vec{\mathcal{A}}$ will be used to represent a wave equation of any arbitrary field and $\vec{\mathcal{J}}_i$ will be used to represent an arbitrary source. 
\begin{align}
\nabla^2\vec{\mathcal{A}}-\mu\varepsilon\frac{\partial^2\vec{\mathcal{A}}}{\partial{t}^2}-\left(\mu\sigma_e+\varepsilon\sigma_m\right)\frac{\partial\vec{\mathcal{A}}}{\partial{t}}-\sigma_m\sigma_e\vec{\mathcal{A}}=-\vec{\mathcal{J}}_i\label{eqn:waveeqn0}
\end{align}
The notation chosen was used to represent vector potentials which will be discussed in a later chapter, but it will equally apply to magnetic and electric fields as long as the source terms are equal (i.e. $\vec{\mathcal{M}}_i=\vec{\mathcal{J}}_i=0$).
